About Parallel and Syntactocentric Formalisms: A Perspective from the Encoding of Convergent Grammar into Abstract Categorial Grammar

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Perspective from the Encoding of Convergent Grammar into Abstract Categorial Grammar

Philippe de Groote, Sylvain Pogodalla, Carl Pollard

To cite this version:

Philippe de Groote, Sylvain Pogodalla, Carl Pollard. About Parallel and Syntactocentric Formalisms:

A Perspective from the Encoding of Convergent Grammar into Abstract Categorial Grammar. Fun- damenta Informaticae, Polskie Towarzystwo Matematyczne, 2011, 106 (2-4), pp.211-231. �10.3233/FI- 2011-384�. �inria-00565598v2�

About Parallel and Syntactocentric Formalisms:

A Perspective from the Encoding of Convergent Grammar into Abstract Categorial Grammar

Philippe de Groote^∗

philippe.degroote@loria.fr LORIA/INRIA Nancy – Grand Est 615 rue du Jardin Botanique 54602 Villers-l`es-Nancy, France

Sylvain Pogodalla^∗C

sylvain.pogodalla@loria.fr LORIA/INRIA Nancy – Grand Est 615 rue du Jardin Botanique 54602 Villers-l`es-Nancy, France

Carl Pollard^∗

pollard@ling.ohio-state.edu The Ohio State University

202 Oxley Hall Columbus OH 43210, United States

Abstract. Recent discussions of grammatical architectures have distinguished two competing ap- proaches to the syntax-semantics interface:syntactocentrism, wherein syntactic structures are map- ped or transduced to semantics (and phonology), vs.parallelism, wherein semantics (and phonology) communicates with syntax via a nondirectional (or relational) interface. This contrast arises for in- stance in dealing within situoperators. The aim of this paper is threefold: first, we show how the essential content of a parallel framework, convergent grammar (CVG), can be encoded within ab- stract categorial grammar (ACG), a generic framework which has mainly been used, until now, to encode syntactocentric architectures. Second, using such a generic framework allows us to relate the mathematical characterization of parallelism in CVG with that of syntactocentrism in mainstream categorial grammar (CG), suggesting that the distinction between parallel and syntactocentric for- malisms is superficial in nature. More generally, it shows how to provide mildly context sensitive languages (MCSL), which are a clearly defined class of languages in terms of ACG, with a rela- tional syntax-semantics interface. Finally, while most of the studies on the generative power of ACG have been related to formal languages, we show that ACG can illuminate a linguistically motivated framework such as CVG.

∗The authors wish to acknowledge support from the Conseil R´egional de Lorraine.

CCorresponding author

Keywords: Grammatical formalism, type theory, linear logic, lambda calculus, mathematics of language, syntax-semantics interface.

Introduction

Analyzing the evolution of generative grammars (GG), [16] uses the termsyntactocentric in reference to grammar formalisms in which the syntactic structures (for instance syntactic proofs/derivations) are mapped or transduced to semantics (and phonology), while advocating a different,parallelarchitecture where semantics (and phonology) communicates with syntax via a nondirectional (or relational) inter- face. Syntactocentric formalisms are exemplified by such frameworks as categorial grammar (CG) [23]

and (though in a weaker sense) principles-and-parameters (P&P), while parallel frameworks include HPSG [29], LFG [19], and Simpler Syntax [7].

The emphasis placed on the nature of the syntax-semantics interface relates to the long-standing challenge for designers of NL grammar frameworks posed byin situ operators, expressions such as quantified noun phrases (QNPs, e.g.every linguist), wh-expressions (e.g. which linguist), and compar- ative phrases (e.g.more than five dollars), whose semantic scope is underdetermined by their syntactic position. One family of approaches, employed by computational semanticists [3] and some versions of CG [1] and phrase structure grammar [6, 29] employs thestoragetechnique first proposed by Cooper [5].

In these approaches, syntactic and semantic derivations proceed in parallel, much as in classical Mon- tague grammar (CMG [21]) except that sentences which differ only with respect to the scope of in-situ operators have identical syntactic derivations.¹ Where they differ is in the semantic derivations: the meaning of an in-situ operator isstoredtogether with a copy of the variable that occupies the hole in a delimited semantic continuation over which the stored operator will scope when it isretrieved; ambiguity arises from nondeterminism with respect to the retrieval site.

Storage is easily grasped on an intuitive level, and its effect can be simulated as in [4], which builds on [22]’s logical reconstruction of [14]. However, the functional mapping between syntax and semantics of this account makes syntactic ambiguity a requirement for semantic ambiguity. Aiming at preserving a relational correspondence between syntactic terms and semantic terms truer to Cooper’s original concep- tion, recent work [28, 27] within the CVG framework provided a partial logical clarification by encoding storage and retrieval rules within a somewhat nonstandard natural-deduction semantic calculus.

In this paper, we first provide a logical characterization of storage/retrieval free of nonstandard fea- tures. To that end, we give an explicit transformation of CVG interface derivations (parallel syntax- semantic derivations) into a framework (ACG [11]) that employs no logical resources beyond those of standard (linear) natural deduction.

Second, by relating the encoding of CVG into ACG to the encoding of CG into ACG as in [26], we show how the modelling of covert movement in the CVGinterfacecalculus strongly relates to the way it is modeled in the so-calledsyntacticcalculus of CG. This underlines the interest of distinguishing between the mathematical apparatus required to achieve some purpose (the encoding of a relation) and the name it is given in different grammatical formalisms (interface or syntax). A generic grammatical framework such as ACG is helpful to make this distinction. Moreover, because the mildly context- sensitive languages (MCSL) are generated by a clearly defined class of ACG, this mathematical apparatus

1In CMG, by contrast, syntactic derivations for different scopings of a sentence differ with respect to the point from which a QNP is ‘lowered’ into the position of a syntactic variable.

can easily be applied to this class of languages and provide them with a parallel architecture, though at first sight they seem inherently syntactocentric in nature.

The remainder of the paper is organized as follows. In Sect. 1, we introduce the basics of CVG.

Section 2 provides a preliminary conversion of CVG by showing how to replace the nonstandard storage and retrieval rules of the semantic calculus by, respectively, standard hypotheses and another rule already present in CVG (analogous to Gazdar’s [10] rule for unbounded dependencies). This conversion requires the addition to the CVG interface calculus of a Shift rule that raises the syntactic type of an in-situ operator to a type similar to that of an ‘overtly moved’ interrogative wh-expression). Section 3 provides an overview of the target framework ACG. In Sect. 4, we lay out the details of the transformation of a (pre-converted) CVG into an ACG and illustrate it with an example. Further examples are given in Sect. 5.Finally, Sect. 6 gives a more general interpretation of our encoding with respect to the way it could be applied to model semantic ambiguity in multiple context-free grammars and with respect to the role of higher-order types.

1. Convergent Grammar

1.1. Introduction

A CVG for an NL consists of three term calculi for syntax, semantics, and the interface. The syntactic cal- culus is a kind of applicative multimodal categorial grammar, the semantic calculus is broadly similar to a standard typed lambda calculus, and the interface calculus recursively specifies which syntax-semantics term pairs belong to the NL.²

1.2. CVG Syntactic Calculus

In thesyntactic calculus, given in Table 1, types are syntactic categories, constants (non-logical axioms) are words (broadly construed to subsume phrasal affixes, including intonationally realized ones), and variables (assumptions) are traces (axiom schema T), corresponding to ‘overt movement’ in generative grammar. Terms are (candidate syntactic analysis of) words and phrases.

For simplicity, we take as our basic syntactic typesnp (noun phrase),s (nontopicalized sentence), andt(topicalized sentence). Flavors of implication correspond not to directionality (as in Lambek calcu- lus) but to grammatical functions. Thus syntactic arguments are explicitly identitifed as subjects (⊸s), complements (⊸c), or hosts of phrasal affixes (⊸a). Additionally, there is a ternary type constructor G (writtenA^C_B) for the category of ‘overtly moved’ phrases that bind anA-trace in aB, resulting in aC.

Contexts (to the left of the ⊢) in syntactic rules represent unbound traces. The elimination rules (flavors of modus ponens) for the implications, also called merges (M), combine ‘heads’ with their syntactic arguments. The elimination rule for the ternary constructor G, inspired by Gazdar’s ([10]) rule for discharging traces³ compiles in the effect of a hypothetical proof step (trace binding) immediately and obligatorily followed by the consumption of the resulting abstract by the ‘overtly moved’ phrase. G

2To handle phonology, ignored here, a fourth calculus would be needed; and then the interface would specify phonol- ogy/syntax/semantics triples.

3With the context corresponding to the value of Gazdar’sSLASHfeature, the major premiss to the “filler” constituent, and the minor premiss to the “gappy” constituent.

requires no introduction rule because it is only introduced by lexical items (‘overt movement triggers’

such as wh-expressions, or the prosodically realized topicalizer).

Table 1. CVG syntactic calculus

⊢a:A Lex t:A⊢t:A ^{T (tfresh)}

Γ⊢b:A⊸sB ∆⊢a:A

Ms

Γ,∆⊢s

a b :B

Γ⊢b:A⊸cB ∆⊢a:A

Mc

Γ,∆⊢ b a^c

:B Γ⊢b:A⊸aB ∆⊢a:A

Ma

Γ,∆⊢ b a^a

:B Γ⊢a:A^C_B t:A; Γ^′⊢b:B

Γ; Γ^′ ⊢a_tb:C G

1.3. CVG Semantic Calculus

In the CVG semantic calculus, given in Table 2, as in familiar semanticλ-calculi, terms correspond to meanings, constants to word meanings, and implication elimination to function application. But just as the CVG syntactic calculus has no introduction rule for the ternary G constructor, correspondingly the semantic calculus lacks the familiar rule for hypothetical proof (λ-abstraction). Instead, binding of semantic variables is effected by either (1) a semantic ‘twin’ of the syntactic G rule, which binds the semantic variable corresponding to a trace by (the meaning of) the ‘overtly moved’ phrase; or (2) by the Responsibility (retrieval) rule (R), which binds the semantic variable that marks the argument position of a stored (‘covertly moved’)in situoperator. Correspondingly, there are two mechanisms for introducing semantic variables into derivations: (1) ordinary hypotheses, which are the semantic counterparts of (‘overt movement’) traces; and the Commitment (Cooper storage) rule (C), which replaces a semantic operatoraof typeA^C_Bwith a variablex :Awhile placinga(subscripted byx) in the store (also called theco-context), written to the left of the⊣(called co-turnstile).

Table 2. CVG semantic calculus

⊢a:A⊣ Lex x:B ⊢x:B ⊣ ^{T (xfresh)}

⊢f :A⊸B ⊣∆ ⊢a:A⊣∆^′

⊢(f a) :B ⊣∆; ∆^′ M

Γ⊢a:A^C_B⊣∆ x:A; Γ^′ ⊢b:B ⊣∆^′ Γ; Γ^′ ⊢a_xb:C ⊣∆; ∆^′ G

⊢a:A^C_B⊣∆

C (xfresh)

⊢x:A⊣a_x :A^C_B; ∆

⊢b:B ⊣a_x:A^C_B; ∆ Γ⊢(axb) :C⊣∆ R

1.4. CVG Interface Calculus

The CVG interface calculus, given Table 3, recursively defines a relation between syntactic and se- mantic terms. Lexical items pair syntactic words with their meanings. Hypotheses pair a trace with a semantic variable and enter the pair into the context. The C rule leaves the syntax of anin situ opera- tor unchanged while storing its meaning in the co-context. The implication elimination rules pair each (subject-, complement-, or affix-)flavored syntactic implication elimination rule with ordinary semantic implication elimination. The G rule simultaneously binds a trace by an ‘overtly moved’ syntactic oper- ator and a semantic variable by the corresponding semantic operator. And the R rule leaves the syntax of the retrieval site unchanged while binding a ‘committed’ semantic variable by the retrieved semantic operator. Example 1.1 demonstrates the use of this R rule in scope ambiguity modelling.

2. About the Commitment and Retrieve Rules

The rules C and R are the only rules of the CVG semantic calculus that manipulate the store (co-context).

Although it is quite clear, in a purely mechanical sense, how these rules are supposed to work, it is considerably less clear how they are to be explicated or justified in terms of well-understood logical notions. In this section we show that, in the presence of a new Shift (syntactic type-raising) rule, they can actually be derived from the other rules, in particular from the G rule.

Proposition 2.1. Any CVG semantic derivationπ can be transformed into a CVG semantic derivation where all C and R pairs of rule have been replaced by the G rule, and which derives the same term.

Proof:

This is proved by induction on the derivations. If the derivation stops on a Lexicon, Trace, Modus Ponens, G or C rule, this is trivial by application of the induction hypothesis.

If the derivation stops on a R rule, the C and R pair has the above schema. Note that nothing can be erased from Γinπ₂ because every variable inΓoccur (freely) only in aand∆. So using a G rule (the only one that can delete material from the left hand side of the sequent) would leave variables in the store that could not be bound later. The same kind of argument shows that nothing can be retrieved from

∆beforeaxhad been retrieved. This means that no R rule can occur inπ₂ whose corresponding C rule is inπ₁(while there can be a R rule with a corresponding C rule introduced inπ₂). Hence we can make the transform and apply the induction hypothesis to the two premises of the new G rule. ⊓⊔ Then, derivations using C and R rules such as the one one on the left can be replaced by the one on the right:

...π₁ Γ⊢a:A^C_B⊣∆ Γ⊢x:A⊣a_x :A^C_B,∆ C

...π₂

Γ,Γ^′ ⊢b:B ⊣a_x:A^C_B,∆^′,∆ Γ,Γ^′ ⊢a_xb:C⊣∆^′,∆ R

...π₁ Γ⊢a:A^C_B⊣∆

x:A⊢x:A⊣ ...π₂

x:A,Γ^′ ⊢b:B ⊣∆^′ Γ,Γ^′ ⊢a_xb:C ⊣∆,∆^′ G

The fact that we can divide the context intoΓandΓ^′ and the store into∆and∆^′, and thatΓand∆are preserved, is a consequence of Proposition 2.1.

Table 3. CVG interface calculus

⊢w, c:A, B⊣ Lex x, t:A, B ⊢x, t:A, B ⊣ ^T Γ⊢f, v:A⊸s B, C⊸D⊣∆ Γ^′ ⊢a, c:A, C ⊣∆^′

Ms

Γ; Γ^′⊢s

a f

,(v c) :B, D⊣∆; ∆^′

Γ⊢f, v:A⊸cB, C ⊸D⊣∆ Γ^′ ⊢a, c:A, C ⊣∆^′

Mc

Γ; Γ^′ ⊢ f a^c

,(v c) :B, C⊣∆; ∆^′

Γ⊢f, v :A⊸aB, C ⊸D⊣∆ Γ^′ ⊢a, c:A, C ⊣∆^′

Mc

Γ; Γ^′ ⊢ f a^a

,(v c) :B, C ⊣∆; ∆^′

Γ⊢a, d:A^C_B, D_E^F ⊣∆ t, x:A, D; Γ^′⊢b, e:B, E ⊣∆^′ Γ; Γ^′⊢a_tb, d_xe:C, F ⊣∆; ∆^′ G

Γ⊢a, b:A, B_C^D ⊣∆

C (xfresh) Γ⊢a, x:A, B⊣b_x:B^D_C; ∆

⊢e, c:E, C ⊣bx :B_C^D; ∆ Γ⊢e,(bxc) :E, D⊣∆ R

Example 1.1. Let’s assume the following CVG lexicon:

liked,like’ :np⊸cnp⊸ss, ι⊸ι⊸π

everyone,ev’ :np, ι^π_π someone,so’ :np, ι^π_π We can build the following derivations:

π_quant^′′ = _⊢liked,like’:np⊸^cnp⊸^ss, ι⊸ι⊸π⊣ ^Lex

⊢someone,so’:np, ι^π_π ⊣ Lex

⊢someone, x:np, ι⊣so’_x:ι^π_π C

⊢

liked someone^c

,like’x:np⊸^ss, ι⊸π⊣so’_x:ι^π_π

π_quant^′ =

...π_quant^′′

⊢

liked someone^c

,like’x:np⊸^ss, ι⊸π⊣so’x:ι^π_π

⊢everyone,ev’:np, ι^π_π⊣ Lex

⊢everyone, y:np, ι⊣ev’y :ι^π_π C

⊢^s

everyone

liked someone^c ,like’x y:s, π⊣so’_x:ι^π_π,ev’_y :ι^π_π Then we can either have:

πSubj. wide scope =

...π_quant^′

⊢^s

everyone

liked someone^c ,like’x y:s, π⊣so’_x:ι^π_π,ev’_y:ι^π_π

⊢^s R

everyone

liked someone^c ,so’_x(like’x y) :s, π⊣ev’_y:ι^π_π

⊢^s R

everyone

liked someone^c ,ev’y(so’x(like’x y)) :s, π⊣ Or:

πObj. wide scope =

...π_quant^′

⊢^s

everyone

liked someone^c ,like’x y:s, π⊣so’x:ι^π_π,ev’y:ι^π_π

⊢^s R

everyone

liked someone^c ,ev’_y(like’x y) :s, π⊣so’_x:ι^π_π

⊢^s R

everyone

liked someone^c ,so’_x(ev’y(like’x y)) :s, π⊣

corresponding to the two readings. Note that the syntactic structure is the same in both cases (s

everyone

liked someone^c ).

This shows we can eliminate the store, resulting in a more traditional presentation of the underlying logical calculus. On the other hand, in the CVG interface calculus, this technique for eliminating C and R rules does not quite go through because the G rule requires both the syntactic type and the semantic type to be of the formα^γ_β. This difficulty is overcome by adding the following Shift rule to the interface calculus:

Γ⊢a, b:A, B_C^D ⊣∆

ShiftE

Γ⊢S_Ea, b:A^E_E, B_C^D ⊣∆

whereSE is a functional term whose application to anAproduces aA^E_E. Then we can transform ...π₁

Γ⊢a, b:A, B_C^D ⊣∆ Γ⊢a, x:A, B⊣bx:B^D_C,∆ C

...π₂

Γ,Γ^′ ⊢e, c:E, C ⊣bx:B^D_C,∆,∆^′ Γ,Γ^′ ⊢e, bxc:E, D⊣∆^′,∆ R to:

...π₁

Γ⊢a, b:A, B_C^D ⊣∆

Shift_E Γ⊢SEa, b:A^E_E, B_C^D ⊣∆

t, x:A, B ⊢t, x:A, B ⊣ ...π₂

t, x:A, B; Γ^′⊢e, c:E, C⊣∆^′ Γ,Γ^′ ⊢(SEa)te, b_xc:E, D⊣∆,∆^′ G

provided(SEa)te= (SEa) (λt.e) =e[t:=a]. This follows fromβ-reduction as long as we takeSE to beλy P.P y. Indeed:

(S_Ea) (λt.e) = (λy P.P y)a(λt.e) =_β (λP.P a) (λt.e) =_β (λt.e)a=_β e[t:=a]

With this additional construct, we can get rid of the C and R rules in the CVG interface calculus. This construct is used in Section 4.3 to encode CVG into ACG. It involves the same rational reconstruction of Montague’s quantifier lowering technique as nothing more thanβ-reduction in the syntax (unavailable to Montague since his syntactic calculus was purely applicative) that was pioneered by [24].

3. Abstract Categorial Grammar

3.1. Motivations

Abstract Categorial Grammars (ACGs) [11], which derive from type-theoretic grammars in the tradi- tion of Lambek [20], Curry [8], and Montague [21], provide a framework in which several grammatical formalisms may be encoded [13]. The definition of an ACG is based on a small set of mathematical prim- itives from type-theory,λ-calculus, and linear logic. These primitives combine via simple composition rules, which offers ACGs a good flexibility. In particular, ACGs generate languages of linearλ-terms, which generalizes both string and tree languages. They also provide the user direct control over the parse structures of the grammar, which allows several grammatical architectures to be defined in terms of ACG.

3.2. Mathematical Preliminaries

LetAbe a finite set of atomic types, and letT_Abe the set of linear functional types types (in notation, α⊸β) built uponA.

Definition 3.1. Given a set of atomic typeA, theordero(α) of a typeαofT_Ais inductively defined as:

• o(α) = 1ifα∈A;

• o(α⊸β) = maxo(β, α+ 1)

Theorderof a typed term is the order of its type.

Definition 3.2. Ahigher-order linear signatureis defined to be a tripleΣ =hA, C, τi, where:

• Ais a finite set of atomic types,

• Cis a finite set of constants,

• andτ is a mapping fromCtoT_A.

A higher-order linear signature will also be called a vocabulary. In the sequel, we will writeA_Σ, C_Σ, andτ_Σto designate the three components of a signatureΣ, and we will writeT_ΣforT_A

Σ.

We take for granted the definition of aλ-term, and we take the relation ofβη-conversion to be the notion of equality betweenλ-terms. Given a higher-order signatureΣ, we writeΛΣfor the set oflinear simply-typedλ-terms.

LetΣandΞbe two higher-order linear signatures. AlexiconL fromΣtoΞ(in notation,L : Σ−→

Ξ) is defined to be a pairL =hη, θisuch that:η is a mapping fromA_Σ intoT_Ξ;θis a mapping from C_Σ intoΛ_Ξ; and for everyc ∈ C_Σ, the following typing judgement is derivable: ⊢_Ξ θ(c) : ˆη(τ_Σ(c)), whereηˆ:T_Σ→T_Ξ is the unique homomorphic extension ofη.⁴

Letθˆ: Λ_Σ → Λ_Ξ be the uniqueλ-term homomorphism that extendsθ.⁵ We will useL to denote bothηˆandθ, the intended meaning being clear from the context. Whenˆ Γdenotes a typing environment

‘x₁:α₁, . . . , x_n:α_n’, we will writeL(Γ)for ‘x₁:L(α₁), . . . , xn:L(αn)’. Using these notations, we have that the last condition forL induces the following property: ifΓ⊢_Σ t :αthenL(Γ)⊢_Ξ L(t) : L(α).

Definition 3.3. Anabstract categorial grammaris a quadrupleG =hΣ,Ξ,L, siwhere:

1. ΣandΞare two higher-order linear signatures, which are called theabstract vocabularyand the object vocabulary, respectively;

2. L : Σ−→Ξis a lexicon from the abstract vocabulary to the object vocabulary;

3. s∈T_Σis a type of the abstract vocabulary, which is called thedistinguished typeof the grammar.

4That isη(a) =ˆ η(a)andη(αˆ ⊸β) = ˆη(α)⊸η(β).ˆ

5That isθ(c) =ˆ θ(c),θ(x) =ˆ x,θ(λx. t) =ˆ λx.θ(t), andˆ θ(t u) = ˆˆ θ(t) ˆθ(u).

A possible intuition behind this definition is that the object vocabulary specifies the surface structures of the grammars, the abstract vocabulary specifies its abstract parse structures, and the lexicon specifies how to map abstract parse structures to surface structures. As for the distinguished type, it plays the same part as the start symbol of phrase structures grammars. This motivates the following definitions.

The abstract languageof an ACG is the set of closed linear λ-terms that are built on the abstract vocabulary, and whose type is the distinguished type:

A(G) ={t∈Λ_Σ| ⊢_Σ t:sis derivable}

On the other hand, theobject languageof the grammar is defined to be the image of its abstract language by the lexicon:

O(G) ={t∈Λ_Ξ| ∃u∈ A(G). t=L(u)}

It is important to note that, from a purely mathematical point of view, there is no structural difference between the abstract and the object vocabulary: both are higher-order signatures. Consequently, the intuition we have given above is only a possible interpretation of the definition, and one may conceive other possible grammatical architectures. One such an architecture consists of two ACGs sharing the same abstract vocabulary, the object vocabulary of the first ACG corresponding to the syntactic structures of the grammar, and that of the second ACG corresponding to the semantic structures of the grammar.

Then, the common abstract vocabulary corresponds to the syntax/semantics interface. This is precisely the architecture that the next section will exemplify.

An equally important notion related to the ACG definition is theorderof an ACG.

Definition 3.4. Theorderof an ACG is the maximum of the order of its abstract constants.

Theorder of thelexicon of an ACG is the maximum of the order of the realizations of its atomic types.

The class of 2nd order ACG has been extensively studied, and an important result is that 2nd order ACG characterize the class of multiple context-free-languages [30, 17, 18] for which polynomial parsing algorithms exist [31].

Terms of the abstract language of a 2nd order ACG cannot have an abstraction as subterm. Hence, they can indeed be seen as trees. With that respect, they are closely related to CVG whereλ-abstraction is forbidden, so that CVG syntactic structures share this property with parse structures of multiple context- free-languages.

4. ACG encoding of CVG

4.1. The Overall Architecture

As Section 1 shows, whether a pair of a syntactic term and a semantic term belongs to the language depends on whether it is derivable from the lexicon in the CVG interface calculus. Such a pair is indeed an(interface) proof termcorresponding to the derivation. So the first step towards the encoding of CVG into ACG is to provide an abstract language that generates the same proof terms as those of the CVG interface. For a given CVGG, we denote byΣ_I(G)the higher-order signature that will generate the same proof terms asG. Then, any ACG whose abstract vocabulary isΣ_I(G)will generate these proof terms.

And indeed we will use two ACG sharing this abstract vocabulary to map the (interface) proof terms into syntactic terms and into semantic terms respectively. So we need two other signatures: one allowing us to express the syntactic terms, which we callΣSimpleSyn(G), and another allowing us to express the semantic terms, which we callΣ_Log(G).

Finally, we need to be able to recover the two components of the pair out of the proof term of the interface calculus. This means having two ACG sharing the same abstract language (the closed terms of Λ(Σ_I(G))of some distinguished type) and whose object vocabularies are respectivelyΣSimpleSyn(G)and Σ_Log(G). Fig. 1 illustrates the architecture withG_Syn=hΣ_I(G),ΣSimpleSyn(G),L_Syn, sithe ACG encod- ing the mapping from interface proof terms to syntactic terms, andG_Sem = hΣ_I(G),Σ_Log(G),L_Log, si the ACG encoding the mapping from interface proof terms to semantic formulas. It should be clear that this architecture can be extended so as to get phonological forms and conventional logical forms (say, in TY₂) using similar techniques. The latter requires non-linearλ-terms, an extension already available to ACG [12] . So we focus here on the (simple) syntax-semantics interface only, which requires only linear terms.

Λ(ΣI(G)) L_Syn

Λ(Σ_SimpleSyn(G)) Λ(Σ_Log(G))

L_Log

G_Syn G_Sem

for instance strings or phonology

Figure 1. Overall architecture of the ACG encoding of a CVG

We begin by providing an example of a CVG lexicon (Table 4). Recall that the syntactic typetis for overtly topicalized sentences, and⊸ ais the flavor of implication for affixation. We recursively define the translation ·^τ of CVG pairs of syntactic and semantics types toΣ_I(G)as:

• α, β^τ = hα, βiif eitherα orβ is atomic or of the formγ_δ^ǫ. Note that this new typeha, βiis an atomictype ofΣ_I(G);

• α⊸β, α^′⊸β^′τ =α, α^′τ ⊸β, β^′τ6.

When inductively ranging over the set of types provided by the CVG lexicon, we get all the atomic types ofΣ_I(G). Then, for anyw, f : α, β of the CVG lexicon ofG, we add the constantw, f^c = W of type α, β^τ to the signatureΣ_I(G).

The application of ·^cand ·^τ to the lexicon of Table 4 yields the signatureΣ_I(G) of Table 5. Be- ing able to use the constants associated to the topicalization operators in building new terms requires additional constants having e.g. hnp, ι^π_πias parameters. We delay this construct to Sect. 4.2.

Table 4. CVG lexicon for topicalization

Chris,C : np, ι top,top’ : np⊸a npts, ι⊸ι^π_π liked,like’: np⊸cnp⊸ss, ι⊸ι⊸π top_in-situ,top’: np⊸a np, ι⊸ι^π_π

6This translation preserves the order of the types. Hence, in the ACG settings, it allows abstraction everywhere. This does not fulfill one of the CVG requirements. However, since it is always possible from an ACGG to build a new ACGG^′such that O(G^′) = {t ∈ A(G)|tconsists only in applications}(it’s enough to transformG intoG^′ as in [15, Chap. 7] whereG^′is a second order ACG), we can assume without loss of generality that we here deal only with second order terms.

Table 5. ACG translation of the CVG lexicon for topicalization CHRIS: hnp, ιi TOP : hnp, ιi⊸hnpts, ι^π_πi

LIKED: hnp, ιi⊸hnp, ιi⊸hs, πi ^TOPIN-SITU: hnp, ιi⊸hnp, ι^π_πi

Constants and types inΣSimpleSyn(G)andΣ_Log(G)simply reflect that we want them to build terms in the syntax and in the semantics respectively. First, note that a term of typeα^γ_β, according to the CVG rules, can be applied to a term of typeα ⊸ β to return a term of typeγ. Moreover, the type α^γ_β does not exist in any of the ACG object vocabularies. Hence we recursively define theJ·Kfunction that turns CVG syntactic and semantic types into linear types (as used in higher-order signatures) as:

• JaK=aifais atomic

• Jα^γ_βK= (JαK⊸JβK)⊸JγK

• Jα⊸xβK=JαK⊸JβK

Then, for any CVG constantw, f :α, βwe havew, f^c=W:α, β^τinΣ_I(G): L_Syn(W) = w L_Log(W) = f

L_Syn(α, β^τ) = JαK L_Log(α, β^τ) = JβK

So the lexicon of Table 4 gives⁷:

L_Syn(CHRIS) =Chris L_Syn(LIKED) =λxy.s

y

liked x^c L_Log(CHRIS) =C L_Log(LIKED) =λxy.like’y x And we get the trivial translations:

L_Syn(LIKEDSANDYCHRIS) =s

Chris

liked Sandy^c :s L_Log(^LIKEDSANDYCHRIS) =like’ C Sandy’:π

4.2. On the Encoding of CVG Rules 4.2.1. Abstraction and Modus Ponens

There is a trivial one-to-one mapping between the CVG rules Lexicon, Trace, and Subject and Com- plement Modus Ponens, and the standard typing rules of linear λ-calculus of ACG: constant typing rule (non logical axiom), identity rule and application. So the ACG derivation that proves ⊢_ΣI(G)

LIKEDSANDYCHRIS:hs, πiinΛ(Σ_I(G)):

⊢^LIKED:hnp⊸np⊸s, ι⊸ι⊸πi cons ⊢SANDY:hnp, ιi ^cons_app

⊢^LIKEDSANDY :hnp⊸s, ι⊸πi ⊢CHRIS:hnp, ιi ^cons_app

⊢LIKEDSANDYCHRIS:hs, πi

7In order to help recognizing the CVG syntactic forms, we use additional operators of arity 2 inΣSimpleSyn(G):s

s p instead of writing(p s)whenpis of typeα⊸sβand

p c^x

instead of just(p c)whenpis of typeα⊸xβwithx6=s. This syntactic sugar is not sufficient to model the different flavors of implication in CVG. While we don’t give the proof here, flavors on implication can be simulated with suitable (flavored) types using for all flavorsfthe translation⌈·⌉fsuch that⌈B⌉f =Bf ifB is an atomic type and⌈β⊸g γ⌉f=⌈β⌉g⊸⌈γ⌉f.

is isomorphic to⊢s

Chris

liked Sandy^c ,like’ Sandy’ C:s, π⊣as a CVG interface derivation:

...π

⊢

liked Sandy^c

,like’ Sandy’:np⊸ss, ι⊸π⊣ ⊢Chris,Chris:np, ι⊣ ^Lex

s app

⊢s

Chris

liked Sandy^c ,like’ Sandy’ C:s, π⊣

whereπ =

⊢liked,like’:np⊸^cnp⊸^ss, ι⊸ι⊸π⊣ Lex ⊢Sandy,Sandy’:np, ι⊣ ^Lex_{c app}

⊢

liked Sandy^c

,like’ Sandy’:np⊸^ss, ι⊸π⊣

But the CVG G rule has no counterpart in the ACG type system. So it needs to be introduced using constants inΣ_I(G).

4.2.2. The G Rule

Let’s assume a CVG derivation using the following rule:

...π₁

Γ⊢a, d:A^C_B, D^F_E ⊣∆

...π₂

t, x:A, D; Γ^′⊢b, e:B, E⊣∆^′ Γ; Γ^′⊢a_tb, d_xe:C, F ⊣∆; ∆^′ G

and that we are able to build two terms (or two ACG derivations) T1 : hA^C_B, D_E^Fi and T2 : B, E^τ of Λ(Σ_I(G))corresponding to the two CVG derivationsπ₁ andπ₂. Then, adding a constantG_hA^C

B,D^F_Ei of typehA^C_B, D^F_Ei⊸(A, D^τ ⊸B, E^τ)⊸C, F^τinΣ_I(G), we can build a new termG_hA^C

B,D^F_Ei^T1(λy.T2) : C, F^τ ∈Λ(Σ_I(G)). The use of this constantG_hA^C

B,D^F_Eiis somehow triggered by a term of typehA^C_B, D^F_Ei (for instance an in-situ operator) and results in a term with a higher-order type reminiscent the one given in CG for the same operators.

It is then up to the lexicons to provide the interpretations ofG_hA^C

B,D_E^Fiso that if:

• L_Syn(T1) =a,

• L_Log(T1) =d,

• L_Syn(T2) =b,

• andL_Log(T2) =e then

• L_Syn(G_hA^C

B,D^F_Ei^T1(λy.T2)) =a(λy.b)

• andL_Log(G_hA^C

B,D^F_Ei^T1(λy.T2)) =d(λy.e).

This is realized whenL_Syn(G_hA^C

B,D^F_Ei) =L_Log(G_hA^C

B,D_E^Fi) =λQ R.Q R.

A CVG derivation using the (not in-situ) topicalization lexical item and the G rule could for instance be⁸:

8With trivial derivations forπSandy topandπChris liked.

...πSandy top

⊢

Sandy top^a

,top’ Sandy’:npts, ι^π_π⊣

...πChris liked

t, x:np, ι⊢^s Chris

likedt^c ,like’xC:s, π⊣

⊢

Sandy top^a (λt.^s

Chris

liked t^c ),(top’ Sandy’)(λx.like’xC) :t, π⊣ This is also isomorphic to the derivation inΛ(Σ_I(G))proving:

⊢_ΣI(G) G

hnpts^,ιππi(TOPSANDY)(λx.LIKEDxCHRIS) :ht, πi. Indeed we have the derivation:

...π^′G Sandy top

⊢G

hnpts^,ιππi(TOPSANDY) : (hnp, ιi⊸hs, πi)⊸ht, πi

...πChris liked^′

⊢λx.LIKEDxCHRIS:hnp, ιi⊸hs, πi

⊢G

hnpts^,ιππi(TOPSANDY)(λx.LIKEDxCHRIS) :ht, πi where:

π^′G Sandy top= ⊢G ^cons

hnpts^,ιππi:hnpts, ι^π_πi⊸(hnp, ιi⊸hs, πi)⊸ht, πi

...π_{Sandy top}^′

⊢TOPSANDY :hnpts, ι^π_πi

⊢G

hnpts^,ιππi(TOPSANDY) : (hnp, ιi⊸hs, πi)⊸ht, πi π^′_{Sandy top}=

⊢^TOP:hnp, ιi⊸hnpts, ι^π_πi cons ⊢SANDY:hnp, ιi ^cons

⊢TOPSANDY:hnpts, ι^π_πi

π^′Chris liked=

⊢LIKED:hnp, ιi⊸hnp, ιi⊸hs, πi cons x:hnp, ιi ⊢x:hnp, ιi ^var_app

x:hnp, ιi ⊢LIKEDx:hnp, ιi⊸hs, πi ⊢CHRIS:hnp, ιi ^cons_app x:hnp, ιi ⊢LIKEDxCHRIS:hs, πi

⊢λx.LIKEDxCHRIS:hnp, ιi⊸hs, πi Let beT=G

hnpts^,ιππi(TOPSANDY)(λx.LIKEDxCHRIS) :ht, πi. Then with

• L_Syn(TOP) =λx.

topx^a

:Jnp⊸a nptsK=np⊸(np⊸s)⊸t,

• L_Log(TOP) =top’:Jι⊸ι^π_πK=ι⊸(ι⊸π)⊸π,

• andL_Syn(G

hnpts^,ιππi) =L_Log(G

hnpts^,ιππi) =λP Q.P Q, we have the expected result:

L_Syn(t) =

Sandy top^a (λx.s

Chris

liked x^c ) L_Log(t) = (top’ Sandy’)(λx.like’xC)

4.3. The C and R Rules

Section 2 shows how we can get rid of the C and R rules in CVG derivations. It brings into play an additional Shift rule and an additional operatorS. It should be clear from the previous section that we could add an abstract constant corresponding to this Shift rule. The main point is that its realization in the syntactic calculus byL_Syn should beS = λe P.P eand its realization in the semantics by L_Log should be the identity.

Technically, this would amount to have a new constantS_hA,B^D

Ci:ha, B_C^Di⊸hA^E_E, B_C^Disuch that:

• L_Log(S_hA,B^D

Ci) =λx.x:JB_C^DK⊸JB_C^DK(this rule does not change the semantics)

• andL_Syn(S_hA,B^D

Ci) =λx P.P x :JAK ⊸(JAK⊸ JEK) ⊸JEK(this rule shifts the syntactic type).

But since this Shift rule is meant to occur together with a G rule to model C and R, the kind of term we will actually consider is: t =G_hA^E

E,B^D_Ci(S_hA,B^D

Cix)Qfor some x :hA, B_C^DiandQ :hA^E_EE, B^D_Ci.

And the interpretations oftin the syntactic and in the semantic calculus are:

L_Log(t) = (λP Q.P Q) L_Syn(t) = (λP Q.P Q)

((λy.y)L_Log(x))L_Log(Q) ((λeP.P e)L_Syn(x))L_Syn(Q)

=L_Log(x)L_Log(Q) =L_Syn(Q)L_Syn(x) So basically,L_Log(λx Q.t) = L_Log(G_hA^E

E,B_C^Di), and this expresses that nothing new happens on the semantic side, whileL_Syn(λx Q.t) =λx Q.Q xexpresses that, somehow, the application is reversed on the syntactic side.

Rather than adding these new constantsS(for each type), we integrate their interpretation into the as- sociatedGconstant⁹. This amounts to compiling the composition of the two terms. So if we have a pair of typeA, B_C^Doccurring in a CVGG, we add toΣ_I(G)a new constantG^S

hA,B^D_Ci:hA, B_C^Di⊸(hA, Bi^τ ⊸ hE, Ci^τ) ⊸ hE, Di^τ (basically the above term t) whose interpretations are: L_Syn(G^S

hA,BC^Di) = λP Q.Q P andL_Syn(G^S

hA,B^D_Ci) =λP Q.P Q.

For instance, if we now use the in-situ topicalizer of Table 4 (prosodically realized by a contrastive pitch accent for instance) we can have the following CVG derivation:

...πSanty top in-situ

⊢Ss

Sandy top_in-situ^a

,top’ Sandy’:npss, ι^π_π⊣

...πChris liked

t, x:np, ι⊢^s Chris

likedt^c ,like’xC:s, π⊣

⊢(Ss

Sandy top_in-situ^a )(λt.^s

Chris

liked t^c ),(top’ Sandy’)(λx.like’xC) :s, π⊣

withπSanty top in-situ=

⊢top_in-situ,top’:np⊸^anp, ι⊸ι^π_π⊣ Lex ⊢Sandy,Sandy’:np, ι⊣ ^Lex

⊢

Sandy top_in-situ^a

,top’ Sandy’:np, ι^π_π⊣

Shifts

⊢Ss

Sandy top_in-situ^a

,top’ Sandy’:npss, ι^π_π⊣ Note that:

(Ss

Sandy top_in-situ^a )t(s

Chris

liked t^c ) = ((λe P.P e)

Sandy top_in-situ^a ) (λt.s

Chris

liked t^c )

=_β s

Chris liked

Sandy top_in-situ^a_c

In order to map this derivation to an ACG term, we use the constant^TOPIN-SITU:hnp, ιi⊸hnp, ι^π_πiand the constant that will simulate the G rule and the Shift rule togetherG^S_hnp,ι^ππi : hnp, ι^π_πi ⊸ (hnp, ιi ⊸ hs, πi)⊸hs, πisuch that, according to what precedes:

• L_Syn(G^S_hnp,ι^ππi) =λP Q.Q P

9It corresponds to the requirement that the Shift rule occurs just before the G rule in modeling the interface C and R rules with the the G rule.

• andL_Log(G^S_hnp,ι^ππi) =λP Q.P Q.

These syntactic and semantic linearizations of in-situ operators are analogous to the one used in [26] to provide a simple syntax to CG.

Then the previous CVG derivation corresponds to the following term ofΛ(Σ_I(G)):

t=G^S_hnp,ι^ππi(^TOPIN-SITUSANDY)(λx.LIKEDxCHRIS)

and its expected realizations as syntactic and semantic terms are:

L_Syn(t) = (λP Q.Q P)(

Sandy top_in-situ^a

) L_Log(t) = (λP Q.P Q)(top’ Sandy’) (λx.s

Chris

liked x^c ) (λx,like’xC)

=s

Chris liked

Sandy top_in-situ^a_c

= (top’ Sandy’)(λx.like’xC) Finally theG_hα,βiandG^S_hα,βiare the only constants of the abstract signature having higher-order types.

Hence, they are the only ones that will possibly trigger abstractions, fulfilling the CVG requirement.

When used in quantifier modeling, ambiguities are dealt with in CVG by the non determinism of the order in which semantic operators are retrieved from the store. This corresponds to the (reverse) order in which their ACG encoding are applied in the final term. However, as they stand, neither account constrains this order. Hence, when several quantifiers occur in the same sentence, all the relative orders of the quantifiers are possible as the next section exemplifies.

5. Examples

5.1. De ReandDe DictoReadings

Quantification is a typical example of covert movement where the semantic scope of QNP is underde- termined by their syntactic position. Incidentally, when two (or more) of these expressions occur in the same sentence, the underdetermination gives rise to several possible relative semantic scope. Table 6 gives a straightforward example of how QNP are encoded in CVG: while the syntactic type is np, as expected, the semantic type using a Gazdar type constructorι^π_π.

Table 6. CVG lexicon for quantifier scope fragment

everyone,ev’ : np, ι^π_π someone,so’: np, ι^π_π thought,think’: s⊸np⊸s, π⊸ι⊸π Kim,K : np, ι Letπ^′′_{de dicto}be the following derivation:

⊢Kim,K:np, ι⊣ Lex

⊢liked,like’:np⊸^cnp⊸^ss, ι⊸ι⊸π⊣ Lex

⊢everyone,ev’:np, ι^π_π⊣ Lex

⊢everyone, x:np, ι⊣ev’x:ι^π_π C

⊢

liked everyone^c

,like’x:np⊸^ss, ι⊸π⊣ev’x:ι^π_π

⊢^s Kim

liked everyone^c ,like’xK:s, π⊣ev’_x:ι^π_π

⊢^s R Kim

liked everyone^c ,(ev’x(like’xK)) :s, π⊣ andπ_{de dicto}^′ be: